# The local–global conjecture for scheduling with non-linear cost

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## Abstract

We consider the classical scheduling problem on a single machine, on which we need to schedule sequentially *n* given jobs. Every job *j* has a processing time \(p_j\) and a priority weight \(w_j\), and for a given schedule a completion time \(C_j\). In this paper, we consider the problem of minimizing the objective value \(\sum _j w_j C_j^\beta \) for some fixed constant \(\beta >0\). This non-linearity is motivated for example by the learning effect of a machine improving its efficiency over time, or by the speed scaling model. For \(\beta =1\), the well-known Smith’s rule that orders job in the non-increasing order of \(w_j/p_j\) gives the optimum schedule. However, for \(\beta \ne 1\), the complexity status of this problem is open. Among other things, a key issue here is that the ordering between a pair of jobs is not well defined, and might depend on where the jobs lie in the schedule and also on the jobs between them. We investigate this question systematically and substantially generalize the previously known results in this direction. These results lead to interesting new dominance properties among schedules which lead to huge speed up in exact algorithms for the problem. An experimental study evaluates the impact of these properties on the exact algorithm A*.

### Keywords

Scheduling Single machine Non-linear cost function Pruning rules Algorithm A*## Notes

### Acknowledgments

We are grateful to the anonymous referees who spotted errors in previous versions of this paper. This paper was supported by the PHC Van Gogh grant 33669TC, the FONDECYT grant 11140566, the NWO grant 639.022.211 and the ERC consolidator grant 617951.

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